Derivatives Pricing. AMSI Workshop, April 2007

Transcription

1 Derivatives Pricing AMSI Workshop, April

2 Overview Derivatives contracts on electricity are traded on the secondary market This seminar aims to: Describe the various standard contracts available in the secondary Expose some pragmatic pricing techniques for these derivatives 2 2

3 Structure Pool price dynamics Forward price dynamics Swaps Swaptions Asians Caps SRAs 3 3

4 $350 P,Q,T ~ ~ ~ $400 Pool Retail supplier Retail supplier Generators 4 4

5 Pool Price Models Stochastic differential equations Diffusion models: ds = a(t,s) dt + b(t,s) dw Standard financial model: GBM ds = S dt + S dw 5 5

6 Diffusion models Do not accurately represent spot price dynamics: 6 6

7 Jump-Diffusion models: ds = a(t,s) dt + b(t,s) dw + K(t,S) dq 7 7

8 MRJD: Standard model Mean reversion supported by physical market environment: Plant flexibility Load response Opportunistic behaviour 8 8

9 Estimating the parameters Step 1: Identify Jumps Method: Recursive filter 1. Evaluate returns ds/s 2. Find threshold of diffusion: 3 stdev(ds/s) 3. Identify outliers as jumps 4. Remove outliers from sample 5. Return to 2 Continue until no new jumps found 9 9

10 Estimating the parameters Step 2: Find volatility, drift and mean Method: Regression 1. ds = (-S)Sdt + S dw 2. Apply Ito: X = log(s); dx = ( - 1/ ) + X]dt + dw Perform best fit by regression: y A + B X dx dx versus X: observations and fit B = dt; - 1/2 2 dt = A; std(y-y*) = dt X 10 10

11 Simulating the process S j+1 = S j + (θ(t j ) log(s j )) S j t + (t j ) Sj N t + K χ( t) Simulated Price Trajectory Time (years) Where N is normal and K is per the jump shape 11

12 Swap derivative Forward contract for electricity Payoff of sold swap: (K - P) V T Hold with physical generation: P G T + (K - P) V T = KVT + P(G-V)T A purely financial cash-settled contract 12 12

13 Swaps Q: What is fair price for entering into a forward contract (swap)? A: What the market is paying: the forward price

14 Observable Forward Prices In Australian secondary market: Quarterly Calendar Peak, Offpeak, Flat Consistency relationships: Flat = peak + offpeak Calendar = Q1 + Q2 + Q3 + Q4 Peak = 7am 10pm on working days Offpeak is everything else 14 14

15 Consistency Q to Cal Period Hours Q Price $/MWh Cal Price $/MWh Q Q Q Q Calendar Price $/MWh Delivery (years) Discount factor Present Value ($/MWh) $250K on 100 MW deal 15 15

16 Comparability of Forwards and Futures Futures requires an initial margin But earn interest at risk-free-rate Futures requires variation margins But could equally be incoming or outgoing Futures can crystalise position now But could be positive or negative Futures provide no credit risk Can be significant 16 16

17 Swaps without visible prices What is fair price for Superpeak Peak-end Shoulderpeak 7-day peak Fully sculptured 17 17

18 Two approaches Method 1 Obtain profile of expected pool prices Transform (scale) profile to fit liquid products Method 2 Find relationship between historical pool outcomes Apply relationship to liquid products 18 18

19 Method 1 example Display of 2004 working day pool prices Mean/median price shape Extract price over 15:44 for the profile Rescale to fit observed peak forward price: F halfhourly = Pprofile P profile / F peak Multiplicative transformation 19 19

20 Kernel smoothing Historical outcomes are influenced by rare events Shown is Q mean pool outcome Events at period 41 are not consistent The event might equally have occurred at period 40 or

21 Method 2 example Historical relationships are remarkably consistent Relationships other than best fit through the origin are justified 21 21

22 Fundamental query What is fundamental relationship between forward and spot prices? F = E(S) Two investigations: What has happened historically? Arguments in risk profiles 22 22

23 Historical Forward-Spot (flat) Anecdotally: Forward price > Spot price Empirically: Flat +$3.5 over data with stdev $15 Quarterly Flat Queensland Forward and Spot Outcomes Resultand spot price Final forward price 23

24 Historical Forward-Spot (peak) Empirically: Peak +$8 over data with stdev $27 Quarterly Peak Queensland Forward and Spot Outcomes Resultant spot price Final forward price 24 24

25 The Risk Premium The arguments: Pool price distributions are asymmetric Forward prices are reflective of spot price plus a risk premium Risk premium may be negative for periods of low pool volatility Risk premium is large for superpeak periods 25 25

26 Risk Premium and Pricing 26 26

27 Swaptions An agreement between two parties that the option holder has the right to enter into a swap at a future date 27 27

28 Option Pricing Classical theory Price a call option on a stock (right to buy) 1973 Black Scholes formula Option payoff $ Share price $ 28 28

29 The Black-Scholes Equation 29 29

30 The Black-Scholes Formula Here N is the standard normal cumulative distribution function

31 Black-Scholes Environment Stock price moves according to a random walk Random changes governed by annualised volatility σ Price increases biased by long-term annual growth µ Increments are uncorrelated Have access to lend/borrow at r 31 31

32 Price ($) Tim e (years) 32 32

33 Price ($) Tim e (years) 33

34 Price ($) µ Tim e (years) 34

35 35 35

36 Suggested pricing method Derive a lognormal probability distribution for the stock price using volatility σ Assume mean growth of µ Calculate expected (mean) payoff NPV at r to arrive at fair value today Not quite right 36 36

37 Delta Hedging 37 37

38 Price ($) r Tim e (years) 38

39 Pricing Options on Swaps Hold a call swaption: right to buy a swap Swaption specifications: Call/put, expiry date, strike price, cash/delivery Underlying specifications: Prevailing forward price, volatility σ, growth µ, interest rate r 39 39

40 Price ($) µ Tim e (years) 40

41 Behaviour of forwards Forward price movements are unbiased (µ = 0). With stocks, µ did not matter delta hedged into r MAJOR DIFFERENCE: - A $35 stock is worth $35 A $35 swap is worth $0 The delta-hedged growth rate is zero 41 41

42 Price ($) µ= Tim e (years) 42

43 Black s formula Similar to Black-Scholes, but assumes zero growth on the underlying Closed form solution (analytic formula) Significant difference? $40/MWh call swaption at 15% volatility over 1.5 years BS = $4.79 Black = $

44 Cash flows and discounting Examine the cash flows of a swaption derivative: $2.68 $

45 Transform from Black-Scholes Put r = 0; Multiply by exp(-r t deliv ) 45 45

46 Nonconstant volatility As delivery approaches, volatility of the forward increases More trading More knowledge about physical Samuelson effect σ = σ 0 + σ 1 exp(- γ(t-t)) Example: Cal 2004 flat 46 46

47 Strategies with swaptions Speculation Covered calls Hedging with a window of speculation Validity periods Large project deals 47 47

48 More on Forward Price Issues 48 48

49 Forward Price Issues Deterministic time-dependent volatility Stochastic volatility 250 day or 365 day Relation with spot Independence of daily increments Volatility estimation methods Unbiased movements Spread in the curve and relation to liquidity 49 49